If that was mine I would gently scrape off any high bumps then use a whole lot of “one time” lite spackle to fill the gaps. Put Blue tape on the stained treads and then gently sand the spackle after it is dry, check with a flashlight to see if there are any spots that need a little more filling, if good then clean the dust off and paint.

Lots of work filling there. Stairs were not finish grade wood. They were meant to be covered with carpeting.

The worst part is the large gap along the top of each tread. Do not try to fix such huge gaps with caulk or spackle: it will look bad and probably end up cracking. If it were me, I'd buy 1/2" quarter round molding, stain it to match the treads, and nail it in place. It would look intentional, rather than like a botched repair job.

First, I would determine if the stairs are structurally sound.

Check the treads to see if they are level. Because it appears like there has been an excessive amount of settling, judging by the gaps on the skirt board.

If the stairs are sound and the treads are level, I would replace the risers that have large gaps and cut a new skirt board.

How far do the stair noses project from the risers? There should be a minimum of 3/4" projection. I'm not sure that yours project a full 3/4" to start with and, if you add vinyl riser facings to the risers, the projection will be even less.

I'm not convinced I see settling. It appears to me that the existing treads could be replacements for older, thicker ones. I base this on the fact that the cutouts on the skirt board for the noses of the treads appear to be larger than needed for the existing treads. Settling would not cause this.

Good point, Kudzu. I wonder how old the structure is.